I don't know about deity worship per se, but elephants are known to practice certain rituals regarding death. They bury their dead, revisit their graves, etc. As far as I know, this is the furthest non-human animals come to religion.

I'd guess the mourning of death and upkeep of various animal graves hints at a belief/concept of spirits, at the very least. Nothing with "self" wants to die and be gone forever, so it would be a pretty natural development of "self" in life forms. It's not as though we humans should expect to be special with our development, and thus stubbornness.

As a child I was told that animals only act on instinct., and I just never believed that...

My dog makes decisions all the time, e.g. I gave her some meat and I didn't realize that my daughter had already given her plenty. She smelled it, walked away, paused, turned around and picked it up. She then went outside and buried it. She had to of thought, I'm not hungry, I better get it and I'll save it for later. I could be wrong, but that's what I think...

Animal brains aren't complex enough to comprehend the idea of a god, or spirits, or anything like that. They do act on instinct, and I think that when animals do things that we could take as being spiritual or showing any type of human like intelligence, its just us anthropomorphizing them. Your dog burying his food may look like he's saving it for later, but hoarding food in order to sustain oneself during lean times is a biological instinct. After all, dogs have no concept of time, and literally have no clue when they are going to eat again.

I don't buy it either. We really don't know that much about the brain and what it can do at various sizes. I think we have a lot more to discover before we can judge what they do and do not think/feel. Many birds use tools and show a "surprising" aptitude for complex problem solving, and we still use the term "bird brain".

Here are just a couple of many articles I found on the subject of animals thinking. I just took a couple of paragraphs to limit the great amount of info.

ScienceDaily (Sep. 15, 2009) — J. David Smith, Ph.D., a comparative psychologist at the University at Buffalo who has conducted extensive studies in animal cognition, says there is growing evidence that animals share functional parallels with human conscious metacognition -- that is, they may share humans' ability to reflect upon, monitor or regulate their states of mind.

The field offers growing evidence that some animals have functional parallels to humans' consciousness and to humans' cognitive self-awareness," he says. Among these species are dolphins and macaque monkeys (an Old World monkey species).

Also...

Rats Capable Of Reflecting On Mental Processes

A new study by researchers from the University of Georgia, just published in the journal Current Biology, shows that laboratory rats do. It's the first demonstration that any non-primate knows when it doesn't know something, and it could open the way to more in-depth studies about how animals--and humans--think."This kind of research may change how we think about cognition and memory in animals," said Jonathon Crystal, an associate professor of psychology in UGA's Franklin College of Arts and Sciences. Crystal's co-author on the paper is Allison Foote, a graduate student in the department of psychology at UGA.

AND

Call me “bird brain”

It has in recent years become clear that we have grossly underestimated the cognitive abilities of birds. Some of the behaviors observed in birds are just as complex, if not more so, than those seen in non-human primates – and “birdbrain” no longer seems so much of an insult.

My parrot tells my husband, SCREW YOU!, only when he puts him back in his cage.

Please don't tell me they don't have a complex thought process because they do.

Ovenbirds Eavesdrop On Chipmunks to Protect Nests

ScienceDaily (June 23, 2011) — Ground-nesting birds face an uphill struggle to successfully rear their young, many eggs and fledglings falling prey to predators. Now, scientists from the USA have found that some birds eavesdrop on their enemies, using this information to find safer spots to build their nests.